# Rayo's number

Rayo's number is a large number named after Mexican philosophy professor Agustín Rayo which has been claimed to be the largest named number.[1][2] It was originally defined in a "big number duel" at MIT on 26 January 2007.[3][4]

## Definition

The Rayo function of a natural number ${\displaystyle n}$ , notated as ${\displaystyle {\mbox{Rayo}}(n)}$ , is the smallest number bigger than every finite number ${\displaystyle m}$  with the following property: there is a formula ${\displaystyle \phi (x_{1})}$  in the language of first-order set-theory (as presented in the definition of ${\displaystyle {\mbox{Sat}}}$ ) with less than ${\displaystyle n}$  symbols and ${\displaystyle x_{1}}$  as its only free variable such that: (a) there is a variable assignment ${\displaystyle s}$  assigning ${\displaystyle m}$  to ${\displaystyle x_{1}}$  such that ${\displaystyle {\mbox{Sat}}([\phi (x_{1})],s)}$ , and (b) for any variable assignment ${\displaystyle t}$ , if ${\displaystyle {\mbox{Sat}}([\phi (x_{1})],t)}$ , then ${\displaystyle t}$  assigns ${\displaystyle m}$  to ${\displaystyle x_{1}}$ . This definition is given by the original definition of Rayo's number.

The definition of Rayo's number is a variation on the definition:[5]

The smallest number bigger than any finite number named by an expression in any language of first-order set theory in which the language uses only a googol symbols or less.

Specifically, an initial version of the definition, which was later clarified, read "The smallest number bigger than any number that can be named by an expression in the language of first-order set-theory with less than a googol (${\displaystyle 10^{100}}$ ) symbols."[4]

The formal definition of the number uses the following second-order formula, where ${\displaystyle [\phi ]}$  is a Gödel-coded formula and ${\displaystyle s}$  is a variable assignment:[5]

{\displaystyle {\begin{aligned}&{\mbox{For all }}R\ \{\\&\{{\mbox{for any (coded) formula }}[\psi ]{\mbox{ and any variable assignment }}t\\&(R([\psi ],t)\leftrightarrow \\&(([\psi ]={\mbox{''}}x_{i}\in x_{j}{\mbox{''}}\land t(x_{i})\in t(x_{j}))\ \lor \\&([\psi ]={\mbox{''}}x_{i}=x_{j}{\mbox{''}}\land t(x_{i})=t(x_{j}))\ \lor \\&([\psi ]={\mbox{''}}(\neg \theta ){\mbox{''}}\land \neg R([\theta ],t))\ \lor \\&([\psi ]={\mbox{''}}(\theta \land \xi ){\mbox{''}}\land R([\theta ],t)\land R([\xi ],t))\ \lor \\&([\psi ]={\mbox{''}}\exists x_{i}\ (\theta ){\mbox{'' and, for some an }}x_{i}{\mbox{-variant }}t'{\mbox{ of }}t,R([\theta ],t'))\\&)\}\rightarrow \\&R([\phi ],s)\}\end{aligned}}}

Given this formula, Rayo's number is defined as:[5]

The smallest number bigger than every finite number ${\displaystyle m}$  with the following property: there is a formula ${\displaystyle \phi (x_{1})}$  in the language of first-order set-theory (as presented in the definition of ${\displaystyle {\mbox{Sat}}}$ ) with less than a googol symbols and ${\displaystyle x_{1}}$  as its only free variable such that: (a) there is a variable assignment ${\displaystyle s}$  assigning ${\displaystyle m}$  to ${\displaystyle x_{1}}$  such that ${\displaystyle {\mbox{Sat}}([\phi (x_{1})],s)}$ , and (b) for any variable assignment ${\displaystyle t}$ , if ${\displaystyle {\mbox{Sat}}([\phi (x_{1})],t)}$ , then ${\displaystyle t}$  assigns ${\displaystyle m}$  to ${\displaystyle x_{1}}$ .

## Explanation

Intuitively, Rayo's number is defined in a formal language, such that:

• ${\displaystyle {\mbox{''}}x_{i}\in x_{j}{\mbox{''}}}$  and ${\displaystyle {\mbox{''}}x_{i}=x_{j}{\mbox{''}}}$  are atomic formulas.
• If ${\displaystyle \theta }$  is a formula, then ${\displaystyle {\mbox{''}}(\neg \theta ){\mbox{''}}}$  is a formula (the negation of ${\displaystyle \theta }$ ).
• If ${\displaystyle \theta }$  and ${\displaystyle \xi }$  are formulas, then ${\displaystyle {\mbox{''}}(\theta \land \xi ){\mbox{''}}}$  is a formula (the conjunction of ${\displaystyle \theta }$  and ${\displaystyle \xi }$ ).
• If ${\displaystyle \theta }$  is a formula, then ${\displaystyle {\mbox{''}}\exists x_{i}(\theta ){\mbox{''}}}$  is a formula (existential quantification).

Notice that it is not allowed to eliminate parentheses. For instance, one must write ${\displaystyle {\mbox{''}}\exists x_{i}((\neg \theta )){\mbox{''}}}$  instead of ${\displaystyle {\mbox{''}}\exists x_{i}(\neg \theta ){\mbox{''}}}$ .

It is possible to express the missing logical connectives in this language. For instance:

• Disjunction: ${\displaystyle {\mbox{''}}(\theta \lor \xi ){\mbox{''}}}$  as ${\displaystyle {\mbox{''}}(\neg ((\neg \theta )\land (\neg \xi ))){\mbox{''}}}$ .
• Implication: ${\displaystyle {\mbox{''}}(\theta \Rightarrow \xi ){\mbox{''}}}$  as ${\displaystyle {\mbox{''}}(\neg (\theta \land (\neg \xi ))){\mbox{''}}}$ .
• Biconditional: ${\displaystyle {\mbox{''}}(\theta \Leftrightarrow \xi ){\mbox{''}}}$  as ${\displaystyle {\mbox{''}}(\neg ((\neg (\theta \land \xi ))\land (\neg ((\neg \theta )\land (\neg \xi ))))){\mbox{''}}}$ .
• Universal quantification: ${\displaystyle {\mbox{''}}\forall x_{i}(\theta ){\mbox{''}}}$  as ${\displaystyle {\mbox{''}}(\neg \exists x_{i}((\neg \theta ))){\mbox{''}}}$ .

The definition concerns formulas in this language that have only one free variable, specifically ${\displaystyle x_{1}}$ . If a formula with length ${\displaystyle n}$  is satisfied iff ${\displaystyle x_{1}}$  is equal to the finite von Neumann ordinal ${\displaystyle k}$ , we say such a formula is a "Rayo string" for ${\displaystyle k}$ , and that ${\displaystyle k}$  is "Rayo-nameable" in ${\displaystyle n}$  symbols. Then, ${\displaystyle {\mbox{Rayo}}(n)}$  is defined as the smallest ${\displaystyle k}$  greater than all numbers Rayo-nameable in at most ${\displaystyle n}$  symbols.

## References

1. ^ "CH. Rayo's Number". The Math Factor Podcast. Retrieved 24 March 2014.
2. ^ Kerr, Josh (7 December 2013). "Name the biggest number contest". Archived from the original on 20 March 2016. Retrieved 27 March 2014.
3. ^ Elga, Adam. "Large Number Championship" (PDF). Archived from the original (PDF) on 14 July 2019. Retrieved 24 March 2014.
4. ^ a b Manzari, Mandana; Nick Semenkovich (31 January 2007). "Profs Duke It Out in Big Number Duel". The Tech. Retrieved 24 March 2014.
5. ^ a b c Rayo, Agustín. "Big Number Duel". Retrieved 24 March 2014.